The born-weight dual boundary element method for linear elastic fracture mechanics

This paper shows that the classical dual boundary element method (DBEM) is one member of a larger family of convex interpolations, the Born-Weight Dual Boundary Element Method (BDBEM), and that this family is the first to reproduce classical DBEM exactly while extending it to the singular spaces that polynomial interpolation cannot span. The boundary integral equations and the exact Kelvin fundamental solution are left unchanged; only the interpolation is reformulated, as convex combinations of dressed nodal values with non-negative weights that sum to unity — the Born weights — so that the partition of unity is built in and the tip element may be given any prescribed singularity exponent, with no node relocated. Two results form the foundation: non-negativity of the weights is incompatible with polynomial reproduction of degree 𝑝 ≥ 2 (Proposition 3), and the obstruction is removed by a Bernstein dressing of the nodal values (Theorem 4) that reproduces any classical polynomial element exactly, thereby placing classical DBEM within the family. Weights built on a fractional power of the intrinsic coordinate span the Williams space; for 𝜆 = 1 2 the tip element is proved equivalent to the quarter-point element, and the stress intensity factors follow in closed form from a single dressed coefficient. The decisive result is an impossibility theorem for the bimaterial interface crack, whose exponent is complex: no placement of nodes can generate the required oscillatory space (Proposition 7), so that the quarter-point construction fails in principle rather than by a small margin—its extracted phase diverges as − ɛ ⁢ l n ⁡ ℓ under mesh refinement, both analytically and numerically. Non-negative Born weights built on the oscillatory Williams functions reach that space directly and recover Rice’s complex stress intensity factor to within 0.17% in modulus and 0.04° in phase; the interface-crack field is thereby spanned exactly, for the first time, without relocating a single node. The method is implemented on classical hardware, verified against the rigid-body and patch-test identities to 1 ⁢ 0 − 1 2 , and assessed on the edge crack, the slant crack, the V-notch, the interface crack, and fatigue growth from a hole. The construction is described most compactly in the language of normalised quantum states, but this language explains the formulation rather than constituting the contribution: the amplitude phases are unobservable, and no quantum computation is involved or required.